Canopus

Canopus is the rank two 3.5.7 temperament tempering out 16875/16807. Having a generator of ~7:5, it possesses non-trivial MOS of the families 1L 2s (triad), 3L 1s (tetrad), 3L 4s ("neutral" diatonic) and 3L 7s (augmented neutral decatonic). On either side the greater region where it appears, there appear the most important, though as yet unnamed, tritave-equivalent temperaments which retain twos, they being important for using a (smeary) ~4:3 or 3:2 as a generator.

The Sigma and Anti-Sigma (Mu) MOS families of 8L+3s and 3L+8s (unfair) or 4L+7s and 7L+4s (fair), but especially the unfair families which by definition include an interval for the function of an "ordinary" ~2:1, are good scales to know for the conceptualizations they provide of how an "ordinary" diatonic or anti-diatonic scale extends into a tritave equivalence. These scales are neighbors of the 7&3 region where the 3L+7s Canopus decatonic scale appears. Below is a list of equal temperaments which contain these scales using generators between or 475.5 and 713.2 cents:

L=1 s=0 8 edt	L=1 s=0 7 edt	L=1 s=0 3 edt
L=7 s=1 59	L=7 s=1 53	L=7 s=1 28
L=6 s=1 51	L=6 s=1 46	L=6 s=1 25
L=5 s=1 43	L=5 s=1 39	L=5 s=1 22
L=4 s=1 35	L=4 s=1 32	L=4 s=1 19
L=7 s=2 62	L=7 s=2 57	L=7 s=2 35
L=3 s=1 27	L=3 s=1 25	L=3 s=1 16
L=5 s=2 46	L=5 s=2 43	L=5 s=2 29
L=7 s=3 65	L=7 s=3 61	L=7 s=3 42
L=2 s=1 19	L=2 s=1 18	L=2 s=1 13
L=7 s=4 68	L=7 s=4 65	L=7 s=4 49
L=5 s=3 49	L=5 s=3 47	L=5 s=3 36
L=3 s=2 30	L=3 s=2 29	L=3 s=2 23
L=7 s=5 71	L=7 s=5 69	L=7 s=5 56
L=4 s=3 41	L=4 s=3 40	L=4 s=3 33
L=5 s=4 52	L=5 s=4 51	L=5 s=4 43
L=6 s=5 63	L=6 s=5 62	L=6 s=5 53
L=7 s=6 74	L=7 s=6 73	L=7 s=6 63
L=1 s=1 11 edt		L=1 s=1 10 edt
L=7 s=6 69	L=7 s=6 70	L=7 s=6 67
L=6 s=5 58	L=6 s=5 59	L=6 s=5 57
L=5 s=4 47	L=5 s=4 48	L=5 s=4 47
L=4 s=3 36	L=4 s=3 37	L=4 s=3 37
L=7 s=5 61	L=7 s=5 63	L=7 s=5 64
L=3 s=2 25	L=3 s=2 26	L=3 s=2 27
L=5 s=3 39	L=5 s=3 41	L=5 s=3 44
L=7 s=4 53	L=7 s=4 56	L=7 s=4 61
L=2 s=1 14	L=2 s=1 15	L=2 s=1 17
L=7 s=3 45	L=7 s=3 49	L=7 s=3 58
L=5 s=2 31	L=5 s=2 30	L=5 s=2 41
L=3 s=1 17	L=3 s=1 19	L=3 s=1 24
L=7 s=2 37	L=7 s=2 42	L=7 s=2 55
L=4 s=1 20	L=4 s=1 23	L=4 s=1 31
L=5 s=1 23	L=5 s=1 27	L=5 s=1 38
L=6 s=1 26	L=6 s=1 31	L=6 s=1 45
L=7 s=1 29	L=7 s=1 35	L=7 s=1 52
L=1 s=0 3 edt	L=1 s=0 4 edt	L=1 s=0 7 edt

As the table shows, the two families overlap at several equal temperaments within the first sixteen proper members of each tree due to the fact that the chain of ~4:3s forms an index-2 subtemperament of a chain of ~3:2s under tritave equivalence. Beyond that, the unfair Sigma and Mu scales match the EDO-EDT correspondences due to their definition including an interval with the function of an "ordinary" ~2:1 which can nevertheless be off by up to +68.0 cents and the fair scales compare to 5a+2b edos in a completely backwards way, with 7L+4s actually comparing to the anti-diatonic scale but being contained in the larger edts. This backward way that the fair scales compare to edos creates an interesting coincidence between 27edt and 27edo both as generated by an ~4:3.

Generator							cents	L	s	notes
3\8							713.23	237.74	0
						22\59	709.20	225.66	32.24
					19\51		708.57	223.76	37.29
						35\94	708.175	222.57	40.47
				16\43			707.74	221.16	44.23
						45\121	707.34	220.06	47.16
					29\78		707.14	219.46	48.77
						42\113	706.92	218.81	50.49
			13\35				706.44	217.37	54.34
						49\132	706.03	216.13	57.635
					36\97		705.88	215.69	58.82
						59\159	705.76	216.32	59.81
				23\62			705.56	214.74	61.35
						56\151	705.36	214.13	62.98
					33\89		705.22	213.70	64.11
						43\116	705.035	213.15	65.585
		10\27					704.43	211.33	70.44
						47\127	703.87	209.66	74.88
					37\100		703.72	209.215	76.08
						64\173	703.61	208.885	76,96
				27\73			703.46	208.43	78.16
						71\192	703.33	208.03	79.25
					44\119		703.24	207.78	79.91
						61\165	703.15	207.49	80.69
			17\46				702.90	206.73	82.69
						58\157	702.63	205.94	84.80
					41\111		702.52	205.62	85.67
						65\176	702.43	205.325	86.45
				24\65			702.26	204.83	87.78
						55\149	702.06	204.24	89.35
					31\84		701.91	203.78	90.57
						38\103	701.69	203.12	92.34
	7\19						700.72	200.21	100.10	Boundary of propriety for unfair Sigma scale
						39\106	699.78	197.37	107.66
					32\87		699.57	196.75	109.31
						57\155	699.43	196.33	110.44
				25\68			699.25	195.71	111.88
						68\185	699.10	195.34	113.09
					43\117		699.01	195.07	113.79
						61\166	698.91	194.78	114.58
			18\49				698.68	194.08	116.45
						65\177	698.46	193.42	118.20
					47\128		698.37	193.17	118.87
						76\207	698.30	192.95	119.45	Golden unfair Sigma scale is near here
				29\79			698.19	192.60	120.38
						69\188	698.05	192.22	121.40
					40\109		697.965	191.94	122.14
						51\139	697.84	191.56	123.15
		11\30					697.38	190.20	126.80
						48\131	696.90	188.74	130.67
					37\101		696.76	188.31	131.82
						63\172	696.65	187.98	132.695
				26\71			696.49	187.52	133.94
						67\183	696.34	187.08	135.11
					41\112		696.25	186.80	135.85
						56\153	696.14	186.47	136.74
			15\41				695.84	185.56	139.17
						49\134	695.49	184.52	141.94
					34\93		695.34	184.06	143.16
						53\145	695.20	183.64	144.29
				19\52			694.945	182.88	146.30
						42\115	694.63	181.93	148.85
					23\63		694.365	181.14	150.95
						27\74	693.96	179.915	154.21
4\11							691.62	172.905		Separatrix of unfair Sigma and Mu scales
						25\69	689.11	192.95	165.39
					21\58		688.64	196.75	163.96
						38\105	688.33	199.25	163.025
				17\47			687.94	202.34	161.87
						47\130	687.63	204.83	160.935
					30\83		687.45	206.24	160.41
						43\119	687.26	207.78	159.83
			13\36				686.82	211.33	158.50
						48\133	686.42	214.51	157.305
					35\97		686.27	215.69	156.86
						57\158	686.15	216.68	156.49
				22\61			685.95	218.26	155.90
						53\147	685.74	219.95	155.26
					31\86		685.59	221.16	154.81
						40\111	685.39	222.75	154.21
		9\25					684.70	228.235	152.16
						41\114	684.04	233.57	150.15
					32\89		683.85	235.07	149.59
						55\153	683.71	236.19	149.17
				23\64			683.515	237.74	148.50
						60\167	683.34	239.17	148.06	Golden unfair Mu scale is near here
					37\103		683.23	240.05	147.725
						51\142	683.10	241.09	147.335
			14\39				682.75	243.84	146.30
						47\131	682.38	246.815	145.19
					33\92		682.22	248.08	144.71
						52\145	682.08	249.22	144.29
				19\53			681.83	251.20	143.54
						43\120	681.53	253.59	142.65
					24\67		681.30	255.49	141.94
						29\81	680.95	258.29	140.89
	5\14						679.27	271.71	135.85	Boundary of propriety for unfair Mu scale
						26\73	677.48	286.60	130.27
					21\59		676.97	290.13	128.95
						37\104	676.66	292.61	128.02
				16\45			676.25	295.86	126.78
						43\121	675.90	298.65	125.75
					27\76		675.695	300.31	125.13
						38\107	675.46	302.18	124.43
			11\31				674.89	306.77	122.71
						39\110	674.33	311.23	121.03
					28\79		674.12	312.98	120.38
						45\127	673.92	314.50	119.81
				17\48			673.61	316.99	118.87
						40\113	673.26	319.80	117.82
					23\65		673.00	321.89	117.04
						29\82	672.64	324.72	115.97
		6\17					671.28	335.64	111.88
						25\71	669.70	348.245	107.15
					19\54		669.21	352.21	105.66
						32\91	668.82	355.31	104.50
				13\37			668.25	359.83	102.81
						33\94	667.71	364.20	101.17
					20\57		667.35	367.04	100.10
						27\77	666.92	370.51	98.80
			7\20				665.68	380.39	95.10
						22\63	664.175	392.37	90.57
					15\43		663.47	398.08	88.46
						23\66	662.80	403.445	86.45
				8\23			661.55	413.47	82.69
						17\49	659.86	426.97	73.63
					9\26		658.37	439.81	73.15
						10\29	655.85	459.09	65.585
1\3							633.985		0
						9\28	611.34	475.49	67.92
					8\25		608.63	456.47	76.08
						15\47	607.01	445.39	80.93
				7\22			605.18	432.26	86.45
						20\63	603.795	422.66	90.57
					13\41		603.06	417.50	92.78
						19\60	602.29	412.09	95.10
			6\19				600.62	400.41	100.11
						23\73	599.25	390.81	104.22
					17\54		598.76	387.425	105.66
						28\89	598.37	384.665	106.85
				11\35			597.76	380.39	108.68
						27\86	597.125	375.97	110.58
					16\51		596.69	372.93	111.88
						21\67	596.135	369.04	113.55
		5\16					594.36	356.62	118.87
						24\77	592.82	345.81	123.50
					19\61		592.41	342.975	124.72
						33\106	592.12	340.92	125.60
				14\45			591.72	338.125	126.80
						37\119	591.36	335.64	127.86
					23\74		591.15	334.13	128.51
						32\103	590.90	332.38	129.26
			9\29				590.26	327.92	131.17
						31\100	589.61	323.33	133.14
					22\71		589.34	321.46	133.94
						35\113	589.10	319.80	134.65
				13\42			588.70	316.99	135.85
						30\97	588.23	313.725	137.25
					17\55		587.88	311.23	138.32
						21\68	587.37	307.67	139.85
	4\13						585.22	292.61	146.30
						23\75	583.27	278.95	152.16
					19\62		582.86	276.09	153.38
						34\111	582.58	274.16	154.21
				15\49			582.23	271.71	155.26
						41\134	581.94	269.68	156.13
					26\85		581.77	268.51	156.63
						37\121	581.59	267.22	157.19
			11\36				581.15	264.16	158.50
						40\131	580.75	261.34	150.71
					29\95		580.60	260.27	160.165
						47\154	580.47	259.36	160.555
				18\59			580.26	257.89	161.18
						43\141	580.03	259.29	161.87
					25\82		579.86	255.14	162.36
						32\105	579.64	253.59	163.025
		7\23					578.86	248.08	165.39
						31\102	578.045	242.41	167.82
					24\79		577.81	240.75	168.53
						41\135	577.63	239.505	169.06
				17\56			577.38	237.74	169.82
						44\145	577.145	236.105	170.52
					27\89		577.00	235.07	170.96
						37\122	576.82	233.85	171.49
			10\33				576.35	230.54	172.905
						33\109	575.82	226.84	174.49
					23\76		575.59	225.23	175.18
						36\119	575.38	223.76	175.81
				13\43			575.01	221.16	176.93
						29\96	574,55	217.93	178.31
					16\53		574.175	215.32	179.43
						19\63	573.605	211.33	181.14
3\10							570.59	190.20
						20\67	567.75	198.72	170.32
					17\57		567.25	200.21	166.84
						31\104	566.93	201.17	164.50
				14\47			566.54	202.34	161.87
						39\131	566.23	203.26	159.71
					25\84		566.06	203.78	158.50
						36\121	565.87	204.34	157.19
			11\37				565.45	205.62	154.21
						41\138	565.07	206.73	151.605
					30\101		564.94	207.14	150.65
						49\165	564.82	207.49	149.85
				19\64			564.64	208.03	148.59
						46\155	564.45	208.60	147.25
					27\91		564.32	209.00	146.30
						35\118	564.14	209.54	145.06
		8\27					563.54	211.33	140.89
						37\125	562.98	213.02	136.94
					29\98		562.82	213.485	135.85
						50\169	562.71	213.83	135.05
				21\71			562.55	214.30	133.94
						55\186	562.41	214.74	132.93
					34\115		562.32	215.00	132.31
						47\159	562.21	215.32	131.58
			13\44				561.94	216.13	129.68
						44\149	561.65	217.00	127.65
					31\105		561.53	217.37	126.80
						49\166	561.42	217.69	126.03
				18\61			561.23	218.26	124.72
						41\139	561.01	218.93	123.15
					23\78		560.83	219.46	121.92
						28\95	560.58	220.23	120.12
	5\17						559.40	223.76	111.88
						27\92	558.18	227.41	103.37
					22\75		557.91	228.235	101.44
						39\133	557.72	228.81	100.10
				17\58			557.47	229.55	98.38
						46\157	557.26	230.17	96.915
					29\99		557.14	230.54	96.06
						41\140	557.00	230.95	95.10
			12\41				556.67	231.95	92.78
						43\147	556.35	232.89	90.57
					31\106		556.23	233.26	89.715
						50\171	556.13	233.57	88.98
				19\65			555.96	234.09	87.78
						45\154	555.77	234.66	86.45
					26\89		555.63	235.07	85.48
						33\113	555.44	235.64	84.16
		7\24					554.74	237.74	79.25
						30\103	553.97	240.05	73.86
					23\79		553.73	240.75	72.23
						39\134	553.55	241.29	70.97
				16\55			553.30	242.07	69.16
						41\141	553.05	242.80	67.445
					25\86		552.89	243.27	66.35
						34\117	552.805	243.84	65.02
			9\31				552.18	245.41	61.35
						29\100	551.57	247.25	57.06
					20\69		551.29	248.08	55.13
						31\107	551.03	248.85	53.33
				11\38			550.57	250.26	50.05
						24\83	549.96	252.07	45.83
					13\45		549.45	253.59	42.27
						15\52	548.64	256.03	36.58
2\7							543.42	271.71	0
						15\53	538.29	251.20	35.89
					13\46		537.51	248.08	41.35
						24\85	537.02	246.135	44.75
				11\39			536.45	243.84	48.77
						31\110	536.00	242.07	51.87
					20\71		535.76	241.09	53.58
						29\103	535.50	240.05	55.40
			9\32				534.925	237.74	59.44
						34\121	534.43	235.78	62.875
					25\89		534.26	235.07	64.11
						41\146	534.11	234.49	65.135
				16\57			533.88	233.57	66.735
						39\139	533.6	232.61	68.42
					23\82		533.475	231.95	69.58
						30\107	533.26	231.09	71.10
		7\25					532.55	228.235	76.08
						33\118	531.90	225.66	80.59
					26\93		531.73	224.96	81.805
						45\161	531.60	224.45	82,69
				19\68			531.43	223.76	83.91
						50\179	531.27	223.13	85.00
					31\111		531.18	222.75	85.67
						43\154	531.065	222.31	86.45
			12\43				530.78	221.16	88.46
						41\147	530.48	218.95	90.57
					29\104		530.35	219.46	91.44
						46\165	530.24	219.01	92.22
				17\61			530.05	218.26	93.54
						39\140	529.83	217.37	95.10
					22\79		529.66	216.68	96.30
						27\97	529.41	215.69	98.04
	5\18						528.32	211.33	105.66	Boundary of propriety for fair Mu scale
						28\101	527.275	207.14	112.99
					23\83		527.05	206.23	114.58
						41\148	526.89	205.62	115.66
				18\65			526.695	204.83	117.04
						49\177	526.53	204.165	118.20
					31\112		526.43	203.78	118.87
						44\159	526.53	203.35	119.62
			13\47				526.07	202.34	121.40
						47\170	525.835	201.38	123.07
					34\123		525.74	201.02	123.70
						55\199	525.67	200.71	124.25	Golden fair Mu scale is near here
				21\76			525.54	200.21	125.13
						50\181	525.40	199.65	126.10
					29\105		525.30	199.25	126.80
						37\134	525.17	198.71	127.74
		8\29					524.68	196.75	131.17
						35\127	524.16	194.69	134.78
					27\98		524.01	194.08	135.85
						46\167	523.89	193.61	136.67
				19\69			523.73	192.95	137.82
						49\178	523.57	192.33	138.91
					30\109		523.47	191.94	139.59
						41\149	523.37	191.47	140.41
			11\40				523.04	190.20	142.65
						36\131	522.675	188.74	145.19
					25\91		522.515	188.105	146.30
						39\142	522.37	187.52	147.335
				14\51			522.105	186.466	149.17
						31\113	521.78	185.15	151.48
					17\62		521.50	184.06	153.38
						20\73	521.08	182.38	156.325
3\11							518.715	172.905		Separatrix of fair Sigma and Mu scales
						19\70	516.24	190.20	163.025
					16\59		512.78	193.42	161.18
						29\107	515.48	195.53	159.98
				13\48			515.11	198.12	158.50
						36\133	514.815	200.21	157.305
					23\85		514.65	201.38	156.63
						33\122	514.46	202.67	155.90
			10\37				514.04	205.62	154.21
						37\137	513.67	208.24	152.71
					27\100		513.53	209.215	152.16
						44\163	513.41	210.03	151.69
				17\63			513.23	211.33	150.95
						41\152	513.03	212.72	150.15
					24\89		512.89	213.70	149.59
						31\115	512.70	215.00	148.85
		7\26					512.59	219.68	146.30
						32\119	511.45	223.76	143.845
					25\93		511.28	224.96	143.16
						43\160	511.10	225.86	142.65
				18\67			510.97	227.10	141.94
						47\175	510.81	228.23	141.29	Golden fair Sigma scale is near here
					29\108		510.71	228.94	140.89
						40\149	510.59	229.77	140.41
			11\41				510.28	231.95	139.17
						37\138	509.94	234.30	137.82
					26\97		509.80	235.29	137.25
						41\153	509.67	236.19	136.71
				15\56			509.45	237.74	135.85
						34\127	509.185	239.62	134.78
					19\71		507.97	241.09	133.94
						23\86	506.66	243.27	132.695
	4\15						507.19	253.59	126.80	Boundary of propriety for fair Sigma scale
						21\79	505.58	264.83	120.38
					17\64		505.21	267.42	118.87
						30\113	504.94	269.30	117.82
				13\49			504.60	271.71	116.45
						35\132	504.30	273.77	115.27
					22\83		504.13	274.98	114.58
						31\117	503.94	265.35	113.79
			9\34				503.46	279.70	111.88
						32\121	503.00	282.935	110.03
					23\87		502.82	284.20	109.31
						37\140	502.66	285.29	108.68
				14\53			502.40	287.09	107.66
						33\125	502.12	289.10	106.51
					19\72		501.90	290.58	105.66
						24\91	501.615	292.61	104.50
		5\19					500.51	300.31	100.10
						21\80	499.26	309.07	95.10
					16\61		498.87	311.80	93.54
						27\103	498.57	313.915	92.33
				11\42			498.13	316.99	90.57
						28\107	497.71	319.955	88.88
					17\65		497.43	321.87	87.78
						23\88	497.10	324.20	86.42
			6\23				496.16	330.775	82.69
						19\73	495.03	338.70	78.16
					13\50		494.51	342.35	76.08
						20\77	494.01	345.81	74.10
				7\27			493.10	352.21	70.44
						15\58	491.885	360.72	65.585
					8\31		490.83	368.12	61.35
						9\35	489.07	380.39	54.34
1\4							475.49		0